String-brane interactions from large to small distancesString-brane interactions from large to small distances Giuseppe D’Appollonio Universit a di Cagliari and INFN DAMTP Cambridge

String-brane interactions from large to small

distances

Giuseppe D’Appollonio

Universita di Cagliari and INFNDAMTP Cambridge

Galileo Galilei Institute, Firenze, 10 April 2019

Giuseppe D’AppollonioString-brane interactions from large to small distances1 / 38

Outline

Strings, high energy and curved spacetimes

The eikonal operator: classical gravity effects and stringycorrections

The eikonal operator from the worldsheet σ-model

String absorption: closed-open transition and the infall of aprobe into a singularity


String backgrounds

String theory from a worldsheet perspective is a perturbativeseries

sum over surfaces or genus espansion

around a given classical background

a (super)conformal σ-model with c = 26 (15)

A good understanding of string compactifications, i.e. vacua ofthe form

R1,d−1 ×KMain examples: toroidal compactifications, orbifolds, compactWess-Zumino-Witten models and their GKO cosets, Gepnermodels.Many lessons: importance of winding states and twisted sectors,T-duality, non-geometric backgrounds...


String backgrounds

Our understanding of curved spacetimes in string theory is morelimited.

Several classes of known examples:

Plane waves (Horowitz and Steif, 1990)

Lorentzian orbifolds (Horowitz and Steif, 1991; Liu, Mooreand Seiberg, 2002)

Non-compact WZW models and their cosets (Witten, 1991;Nappi and Witten, 1992 and 1993)

Change of strategy: analyze the high-energy dynamics of a stringin the background of a collection of D-branes.


String-brane system at high energy

The string-brane system provides an excellent framework for thestudy of string dynamics in curved spacetimes

Microscopic description at weak coupling in terms of openstrings

Geometric description at strong coupling, extremal p-branes

Unitary S-matrix

The high-energy limit makes the dynamics both interesting and(to a certain extent...) tractable

The large energy causes an enhancement of classical andquantum gravity effects

The large energy reveals the full string dynamics:interactions of states of arbitrary mass and spin. Inelasticprocesses grows in variety and importance.


Extremal p-branes

Low energy effective action in the string frame

S =1

(2π)7l8s

∫d10x√−g[e−2ϕ

(R + 4(∇ϕ)2

)− 2

(8− p)!F 2p+2

].

BPS solutions carrying R-R charges Horowitz and Strominger, 1991

ds2 =1√H(r)

ηµν dxµdxν +

√H(r) δij dx

idxj , eϕ = gH3−p4 ,

∫S8−p

∗Fp+2 = N , H(r) = 1 +

(R

r

)7−p,

R7−p

l7−ps

= dp gN ,

λ = gN , dp = (4π)5−p2 Γ

(7− p

2

), τp =

R7−p

(2π)7dpg2α′4 .


String-brane collisions



Relevant scales

α′s 1 ,

(R

ls

)7−p

∼ g N , b8−pT ∼ α′ER7−p , bc ∼ R

Various possible processes as the impact parameter is varied

b bT R elastic scattering

bT ≥ b R string tidal excitations

b < bc

R ls creation of open strings

R ls infall into the singularity

Fixed effective background: extremal p-brane metric




The eikonal operator

Regge limit of the disk amplitude (tree level)

A1(s, t) ∼ Γ

(−α

′

4t

)e−iπ

α′t4 (α′s)1+

α′t4 ,

s = E2 , t = −(p1 + p2)2

Grows too fast with energy. Include higher-orders.



The result is the eikonal operator

S(s, b) = e2iδ(s,b) ,

2δ(s, b) =

2π∫0

dσ

2π

: A1 (s,b + X(σ)) :

2E

Amati, Ciafaloni e Veneziano (1987)GD, Di Vecchia, Russo e Veneziano (2010).

In impact parameter space the tree-level amplitude is

A1(s, b) ∼ s√π

Γ(6−p2

)Γ(7−p2

)R7−pp

b6−p+

iπ√s

Γ(7−p2

)√ πα′s

lnα′s

(Rpls(s)

)7−pe− b2

l2s(s)

ls(s) is the effective string length ls(s) = ls√

lnα′s



Two main effects

deflection of the trajectory

excitation of the internal degrees of freedom of the string:tidal forces

Scattering angle θ = − 2E∂δ(s,b)∂b

The leading and next-to-leading terms

Θp =√π

[Γ(8−p2

)Γ(7−p2

) (Rp

b

)7−p

+1

2

Γ(15−2p

2

)Γ (6− p)

(Rp

b

)2(7−p)

+ . . .

]

are in perfect agreement with the classical deflection of a masslesspoint-like probe in the p-brane background.


Tidal excitation at leading order

When b R ls√

ln(α′s)

2 δ(s,b + X) ∼ 1

2E

[A1(s, b) +

1

2

∂2A1(s, b)

∂bi∂bjX iXj + ...

]where Q ≡ 1

2π

∫ 2π

0dσ : Q(σ) : and the string coordinates are

X i = i

√α′

2

∑n6=0

(Ainn

einσ +Ainne−inσ

), [Ain, A

jm] = nδijδn+m,0

Matrix of the second derivatives of the eikonal phase

1

4√s

∂2A1(s, b)

∂bi∂bj= Q⊥(s, b)

[δij −

bibjb2

]+ Q‖(s, b)

bibjb2

where

Q⊥(s, b) =1

4√s

1

b

dA1(s, b)

db, Q‖(s, b) =

1

4√s

d2A1(s, b)

db2


Tidal excitation at leading order

At leading order in Rb

Q⊥(s, b) = −√π

2

√s

Γ(8−p2

)Γ(7−p2

)R7−p

b8−p, Q‖(s, b) = −(7− p) Q⊥(s, b)

String corrections contribute a non-vanishing imaginary part tothe eikonal phase

∣∣∣〈0|e2iδ(s,b)|0〉∣∣∣ ∼ e− 1

2√sImA1(s,b) (2πα′|Q⊥|)

8−p2√

7− p e−πα′(7−p)|Q⊥|

The absorption of the elastic channel due to string excitationsbecomes non negligible for b ≤ bT

b8−pT =π

2α′√πs(7− p)

Γ(8−p2

)Γ(7−p2

)R7−pp


Tidal excitation and the pp-wave limit

Can we reproduce these results starting with the sigma-model for theextremal p-brane metric?

ds2 = α(r)

(−dt2 +

p∑a=1

(dxa)2

)+ β(r)

(dr2 + r2(dθ2 + sin2 θdΩ2

7−p))

where β(r) = 1/α(r) =√H(r).

We focus on a small neighborhood around a null geodesic expandingthe sigma model action in Fermi coordinates. The leading term inenergy corresponds to the Penrose limit of the brane background

ds2 = 2dudv +

p∑a=1

dx2a +

7−p∑i=1

dy2i + dy20 + G(u, xa, yi, y0)du2 ,

G =∂2u√α√α

p∑a=1

x2a +∂2u(√βr sin θ)√βr sin θ

7−p∑i=1

y2i +∂2u√βr2 − b2α√βr2 − b2α

y20 .

Blau, Figueroa-O’Farrill and Papadopoulos, 2002Giuseppe D’Appollonio

String-brane interactions from large to small distances15 / 38

The bosonic part of the string sigma model becomes

S ∼ S0 −1

4πα′

∫dτ

∫ 2π

0dσ ηαβ ∂αU∂βU G(U,Xa, Y i, Y 0)

where S0 is the string action in Minkowski space.We work in the light-cone gauge U(σ, τ) = α′Eτ and evaluate thetransition amplitudes in the impulsive approximation. The integralsover u then decouple from the string coordinates and simply providec-number coefficients to the quadratic action of the fluctuations

S − S0 ∼E

2

∫ 2π

0

dσ

2π

(cx

p∑a=1

X2a(σ, 0) + cy

7−p∑i=1

Y 2i (σ, 0) + c0Y

20 (σ, 0)

)

cy =

∫ +∞

−∞duGy(u) = 2

∫ ∞0

∂2u(√βr sin θ)√βr sin θ

=⇒ E

2cy = Q⊥(s, b)

c0 =

∫ +∞

−∞duG0(u) = 2

∫ ∞0

∂2u√βr2 − b2α√βr2 − b2α

=⇒ E

2c0 = Q‖(s, b)


High-energy limit of the string propagator

Working in Fermi coordinates it is possible to establish a precisedictionary to compare the amplitudes derived from the σ-model inthe curved background and those obatined by the eikonalresummation of string amplitudes in flat space in the presence ofD-branes.

It is also possible to include higher-order terms in the stringcoordinates, adapting to Fermi coordinates a recursive algorithmdeveloped by Sunil Mukhi (1986) for the expansion in Riemanncoordinates.

Using Mukhi’s method one can for instance derive the full leadingeikonal operator (i.e. including all the corrections in α′) from theworldsheet σ-model.

Work in progress with Alessio Caddeo.Giuseppe D’Appollonio

String-brane interactions from large to small distances17 / 38


Existence and form of the eikonal operator deduced from theelastic amplitude. An inclusive sum over the intermediate states.

Natural interpretation: Hilbert space of the string quantized in alight-cone gauge aligned to the collision axis.

This can be proved in two different ways

Light-cone derivation: Regge limit of the three-string vertexof Green and Schwarz

Covariant derivation: Reggeon vertex operator

GD, Di Vecchia, Russo and Veneziano, 2013


The Reggeon vertex operator

Elegant description in terms of an effective string state: theReggeon

A(s, t) ∼ ΠDpR C12R C12R .

Factorized form for the four (two) point amplitudes

Process independent propagator (tadpole)

Evaluation of three-point couplings

Reggeon tadpole

ΠDpR = A1(s, t) =

π9−p2

Γ(7−p2

)R7−pp Γ

(−α

′t

4

)e−iπ

α′t4 (α′s)1+

α′t4

Three-point coupling with the Reggeon

CS1,S2,R =⟨V

(−1)S1

V(0)S2V

(−1)R

⟩= εµ1...µr ζν1...νs T

µ1...µr;ν1...νsS1,S2,R


The Reggeon vertex operator

The Reggeon vertex is a superconformal primary of dimension onehalf in the high-energy limit Ademollo, Bellini, Ciafaloni (1989)

Brower, Polchinski, Strassler, Tan (2006)

Reggeon vertex operator (picture (−1))

V(−1)R =

ψ+

√α′E

(√2

α′i∂X+

√α′E

)α′t4

e−iqX .

Reggeon vertex operator (picture (0))

V(0)R =

[− 2

α′∂X+∂X+

α′E2− iqψψ

+∂X+

α′E2− α′t

4

ψ+∂ψ+

α′E2

](√

2

α′i∂X+

√α′E

)α′t4−1

e−iqXL


Absorption of closed strings by D-branes

In the background of a Dp-brane a closed string can split turningitself into an open string.Initial closed string at level Nc

pc = (E,~0p, ~pc) , α′M2 = 4(Nc − 1) .

Final open string at level n

po = (−m,~0p,~025−p) , α′m2 = n− 1 .

The transition is possible if

α′~p 2c = n− 4Nc + 3 ≥ 0 .

The Chan-Paton factor corresponds to the U(1) in thedecomposition U(N) = U(1)× SU(N).


The closed-open vertex

The closed-open light-cone vertex describes the transition from anarbitrary closed string to an arbitrary open string

Cremmer and Scherk (1972), Clavelli and Shapiro (1973), Green and Schwarz (1984), Shapiro and Thorn(1987), Green and Wai (1994).

Same ideas and techniques used to derive the vertex for three openstrings

Path integral (Mandelstam, 1973)

Operator solution of the continuity conditions for the stringcoordinates and supersymmetry(Green and Schwarz,1983)

Calculation of the three-point couplings of DDF operators(Ademollo, Del Giudice, Di Vecchia and Fubini, 1974)

Historically, showing that Mandelstam ↔ ADDF was important torealize that

Dual models ↔ Theory of interacting relativistic strings



Chosen two light-cone directions e±, the vertex is given by

|VB〉 = β exp

3∑r,s=1

∞∑k,l=1

1

2Ar,i−kN

rskl A

s,i−l +

3∑r=1

∞∑k=1

P iN rkA

r,i−k

3∏r=1

|0〉(r) .

The index i runs along the d− 2 directions orthogonal to e±

The index r = 1, 2→ left, right parts of the closed stringThe index r = 3→ open string

p(1) =pc2, p(2) =

Dpc2

, p(3) = po ,

pc = (E,~0p, ~pt, p) , po = (−m,~0p,~024−p, 0) .

A1,ik = Aik , A2,i

k = (DAi)k , A3,ik = aik .



The quantities αr are given by

αr = 2√

2α′(e+p(r)) , r = 1, 2, 3 , α1 + α2 + α3 = 0 .

We also have

Pi ≡√

2α′[αrp

(r+1)i − αr+1p

(r)i

].

The Neumann matrices N in the vertex are

N rk = − 1

kαr+1

(−kαr+1

αr

k

)=

1

αrk!

Γ(−kαr+1

αr

)Γ(−kαr+1

αr+ 1− k

) ,

N rskl = − klα1α2α3

kαs + lαrN rkN

sl .



Light-cone gauge lying in the branes worldvolume (p ≥ 1)

e+ =1√2

(−1, 1, . . . , 0, 0) , e− =1√2

(1, 1, . . . , 0, 0) .

We find

α1 = α2 =√α′E , α3 = −2

√α′E ,

and

Pi = α3

√α′

2pc,i .

This is the form usually displayed in the literature.



Light-cone gauge along the direction of collision

e+ =1√2

(−1, 0, . . . , 0, 1) , e− =1√2

(1, 0, . . . , 0, 1) .

We find

α1 =√α′ (E + p) =

√n− 1 +

√n− 1− 4ω ,

α2 =√α′ (E − p) =

√n− 1−

√n− 1− 4ω ,

α3 = −2√α′E = −2

√n− 1 ,

where

ω =α′

4(M2 + ~pt

2) = Nc − 1 +α′~pt

2

4.



Closed-open couplings and discontinuity from the vertex

Bψχ = 〈χ|V~pt|ψ〉 , ImAψψ(s, t) = πα′〈ψ2|V †~q/2V~q/2|ψ1〉 .

At high energy (large level limit)

V~pt ∼ V~pt , n 1 .

Bψχ = 〈χ|V~pt |ψ〉 , ImAψψ(s, t) = πα′〈ψ|V†~0V~q|ψ〉 .

In impact parameter space

V~b =

∫d24−pq

(2π)24−pe−i

~b~q V~q .

Bψχ(~b) = 〈χ|V~b|ψ〉 , ImAψψ(s,~b) = πα′〈ψ|V†0V~b|ψ〉 .



When n tends to infinity, the coefficients quadratic in the openmodes behave like

N33kl ∼ −ω

n

1

k + l

k−ωkn

Γ(1− ω k

n

) l−ωln

Γ(1− ω l

n

) .If the ratios k/n and l/n tend to zero this is

N33kl ∼ −

ω

n

1

k + l.

If k and l are of order n, setting k = nx and l = ny we find

N33kl ∼ −n−ω(x+y)−2 x−ωxy−ωy

Γ(1− ωx)Γ(1− ωy)

ω

x+ y.

The coefficients N13kl (coupling the open modes and the left modes

of the closed string) are enhanced by an additional power of nwhen k = l.


Absorption of a tachyon

Let us start from the absorption of a tachyon with ~pt = 0

V~0|0〉 = β Pn e12

∑k,lN

33kl a

i−ka

i−l |0〉 .

Series representations for the open state

V~0|0〉 = β∞∑Q=0

1

Q!2QPn

Q∏α=1

∑kα,lα

N33kαlαa

iα−kαa

iα−lα|0〉 ,

and for the imaginary part of the elastic amplitude

ImATT = πα′β2

∞∑Q=0

1

(Q!)222Q

∑kα,lα

Q∏α=1

(N33kαlα

)2〈0|

Q∏α=1

aiαkαaiαlα

Q∏α=1

aiα−kαaiα−lα|0〉 ,

Q∑α=1

(kα + lα) = n .



The first few terms give an accurate representation of the state

ImATT ∼ πα′β2∑k,l

k+l=n

1

4

(N33kl

)2 〈0|aikailaj−kaj−l|0〉= πα′β2

∑k,l

k+l=n

12(N33kl

)2kl ,

ImATT ∼ 12 πα′β2 n

∫ 1

0

dx

∫ 1

0

dyx2x+1y2y+1δ(x+ y − 1)

Γ2(1 + x)Γ2(1 + y).

We findImATT ∼ 0.929πα′β2 n .



Including terms with four oscillators we find

ImATT ∼ 0.998πα′β2 n .

Therefore only 0.2% of the full forward imaginary part is left toterms with six or more oscillators. This is strong evidence thatthe series converges rapidly and that

〈0|eZ†0PneZ0|0〉 = n .


Classical solutions

Intuitive picture of the absorption process: the closed stringhits the branes and it is turned into an open string thatstretches along the direction of motion of the closed string.

Since the open string cannot have a momentum zero-mode ina direction orthogonal to the branes, the momentum p of theclosed string is turned into open string excitations (these cancarry, at any given time, a net transverse momentum)

The string performs a periodic motion converting, as itstretches and contracts, its kinetic energy into tensile energyand vice versa.


Classical solutions

It is convenient to choose the same range 0 ≤ σ ≤ π for theworldsheet coordinate σ of the closed and the open string andwork in a static gauge.

We will assume that the closed-open transition happens atτ = 0 and require continuity of the string coordinates andtheir first time derivatives at τ = 0.

We denote with Xa, a = 1, ..., p the directions parallel to thebranes, with Z = X25 the direction of collision and with X i,i = p+ 1, ..., 24 the remaining coordinates orthogonal to thebranes.

The position of the branes in the transverse directions isX i = Z = 0.

Without loss of generality we can choose a frame where theonly non vanishing components of the closed momentum arep0 = E and pZ = p.


Classical solutions

Head-on collision at zero impact parameter of a closed stringwithout excitations

X0 = 2α′Eτ , Z = 2α′Eτ , τ < 0 .

The open string solution at τ > 0 is

X0 = 2α′Eτ , Z = α′E (W (τ + σ)−W (τ − σ)) ,

where W (ξ) is a triangular wave with period 2π

W (ξ) = ξ , ξ ∈ [0, π] , W (ξ) = 2π − ξ , ξ ∈ [π, 2π] ,

whose Fourier series is

W (ξ) =π

2− 4

π

∞∑k=0

1

(2k + 1)2cos(2k + 1)ξ .


Classical solutions

The open string undergoes a periodic motion in the Z directionwith period 2π. When τ ∈ [0, π/2]

Z(σ, τ) =

2α′Eσ 0 ≤ σ ≤ τ

2α′Eτ τ ≤ σ ≤ π − τ2α′E(π − σ) π − τ ≤ σ ≤ π

.

The sum (Z)2 + (Z ′)2 remains constant all along the open stringand equal to (α′E)2.At a generic positive τ < π/2 the points of the string at 0 < σ < τand π − τ < σ < π have stretching energy (Z ′) and no kineticenergy (Z).The complementary points at τ < σ < π − τ have kinetic but nostretching energy.


Classical solutions

When τ = π/2 all the energy of the string is due to its stretchingas it reaches the maximum extension in the Z direction,Zmax = πα′E.The average length of the string over a period of oscillation is

〈L〉 =1

2π

∫ 2π

0

dτ

∫ π

0

dσ

√(∂σZ)2 = πα′E .

The mean square distance of the points of the string from theposition of the branes is⟨

∆Z2⟩

=1

2π2

∫ 2π

0

dτ

∫ π

0

dσ Z2(τ, σ) =π2

6α′

2E2 .

The way a D-brane absorbs an arbitrary incident energy is byconverting it into the length of its open string excitations (at treelevel).


Classical solutions

T-dual picture of the solution: both the branes and the initialclosed string are wrapped around a circle. The initial energy isentirely due to the stretching of the closed string

E =wR

α′.

At τ = 0 the closed string splits into an open string satisfyingNeumann boundary conditions

Z = α′E (W (τ + σ) +W (τ − σ)) .

The T-dual picture makes it intuitive why the total decay rate

Γ =ImA2E∼ E ,

grows linearly with the energy: it simply reflects the constantsplitting probability per unit length of the closed string.


Conclusions

The string-brane system provides a convenient framework toaddress several problems in quantum gravity and to study thestructure and symmetries of string theory

emergence of an effective geometry from the scattering dataexistence of a unitary S-matrix for high-energy collisionsmicroscopic description of the infall of a particle into asingularity

Some work in progress

study less symmetric backgrounds (e.g. intersecting branes)study the open string state created by the absorption of theclosed string at weak couplingderive a general formula for the phase shift in anasymptotically flat spacetimederive the full structure of the eikonal operator at the nextorder in R7−p

b7−p


Documents

String-brane interactions from large to small distancesString-brane interactions from large to small distances Giuseppe D’Appollonio Universit a di Cagliari and INFN DAMTP Cambridge